My guess would be: by parts.
Here you'd set u = x^2, dv = sinh mx
You'll have an expression in x and sinh mx (or cosh mx) at the end of that, then you'll use parts again, this time with u=x.
I tried integrating by parts with u=sinh (mx) and dV= x^2. Now I have
sinh [mx]/3 - (1/3) integral x^3cosh [mx].
So I tried integrating by parts again, using u= x^3, but it just makes it more complicated.
And sorry my posts are hard to read. Not sure how to use the tags for these.